"Wysłałem ci list."

Translation: I have sent you a letter.

[Walkinthedog]

Most often here we would drop the have, I sent you a letter is pretty much the same as i have sent

[BenYoung84]

I don't think have is any more omitable in this case than in any other.

[BorisMiede]

I have sent a letter to you

[Jellei]

Works.